Stumbling and Mumbling

Author: chris dillow   |   Latest post: Mon, 19 Apr 2021, 4:26 PM


A defence of maths in economics

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What's the point of maths in economics? I ask because of this:

Much of the mathematics used in economics is simply performance. Despite the sometimes undue attention given to mathematical methods, the actual mathematics economists use often contradicts their economic conclusions. The mathematics in economics is simply a tool to keep out outsiders: economists themselves do not take it that seriously.

I suspect this is sometimes (often?) true. But I want to give a counter-example. It comes from a source some of you don't like, John Cochrane. It's this equation (pdf) for predicting long-run returns on assets relative to returns on cash:

ER = RA x SDa x SDbg x corr (a, bg)

Where RA is a coefficient of risk aversion, SDa is the standard deviation of asset returns, SDbg is the volatility of background risk, and corr(a, bg) is the correlation between asset returns and background risk. The intuition here is simple. High expected returns should be compensation for higher risk or for a higher distaste for risk. And the risk that matters isn't just the volatility of the asset, but its correlation with our background risks - the risks to our job or business. An asset that falls when we lose our job or business is much riskier than one that holds up well in bad times, and so must pay us higher returns on average to compensate us.

Now, the derivation of this is quite tricky: see the appendix to Cochrane's paper. But once something has been derived once, it stays derived. Working through the derivations might be useful for torturing students, but the rest of us can skip them and get to the meat.

We can apply this equation to any asset. I suspect it has done a decent job of explaining long-run returns on UK housing, for example.

Let's put some numbers on this for equities. The coefficient of risk aversion varies across individuals, across time, and from context to context, but let's call it three (pdf), where one betokens indifference to risk. The standard deviation of annual equity returns has been 15% (or 0.15) since 1985. The volatility of background risk - how likely we are to lose our job or business - is hard to measure and varies from person to person, but let's call it 0.1 for illustrative purposes. Corr (a, bg) also varies from person to person. For most of us, though, it's positive: we know this because shares fall in recessions. Let's call it 0.4.

We can then multiply these numbers through: 3 x 0.15 x 0.1 x 0.4 = 0.018. Which tells us we should expect equities to out-perform safe assets by 1.8 percentage points a year on average over the long-term. Eqpm

My chart shows how this compares to reality. It shows that unless you had bought at the right time (such as after the bursting of the tech bubble or at the low point of the 2009 crisis) you'd have earned an equity premium of less than this on the All-share index. For the MSCI world index, however, the equity premium for pretty much any period in the 90s was very close to this prediction. Thanks to the boom in US equities, though, the premium has been more than this for most of this century.

Which represents a return to history. Back in 1985 Rajnish Mehra and Ed Prescott pointed out - based on this sort of thinking - that equities had done better (pdf) than they should for much of the 20th century.

Why? Herein lies one virtue of this equation. It is sufficiently clear and parsimonious for us to ask: what does it leave out?

One thing is disaster risk. The (not so small) risk of a catastrophe (pdf) justified higher expected returns on equities for much of the 20th century - especially for investors who were loss-averse. Another thing is market imperfections. Roger Farmer has pointed out (pdf) that many of the young are excluded from stock markets by borrowing constraints. That causes shares to be cheaper than they should be and riskier too - because such people cannot buy when prices fall. Which justifies higher returns.

I'd highlight several merits of this equation - ones which give us criteria by which to assess when maths is useful in economics.

First, it is simple enough to be useful. If we need Matlab to solve equations, they are too complicated for us to fully understand and manipulate. The maths then becomes "a tool to keep out outsiders", and to prevent them following what's going on.

Secondly, this equation allows reverse engineering. If people expect high returns, we can ask: what terms on the right hand side of the equation justify this? We need high expected returns on Bitcoin, for example, because the thing is so damned volatile and - as its fall last March showed - correlated with background risk.

Thirdly, maths can be useful without precise numbers - as my illustration showed. As the late Thomas Mayer said, there's a trade-off between truth and precision. Equations are ways of organizing our thinking. They allow us to ask: what is it that we need to know? What might be missing from this equation? But it is only equations that are simple that allow us to do this.

Finally, this equation is useful for individuals. Economics should not be confined to academia, nor is it a subsidiary of politics. It should instead help people in their everyday life. Which is what this equation does. It draws our attention to an important fact. If you have low background risk - say because you're in a safe job or have retired on a big final salary pension - you are better able to take on risks which are correlated with that risk than others. Assets that fall in recessions are less risky for you than they are for people who might lose their jobs in such recessions. You can therefore pick up a risk premium in normal times for taking on a risk that doesn't bother you. Which means you're getting something for nothing.

Now, imagine you retire from a job that you could lose in a downturn and take a final salary pension. Your background risk falls. So you're better placed to take the recession risk which equities carry. Which means that you should own more shares when you are old than when you were younger. For you, the conventional financial advice that people should own fewer shares as they age is flat wrong. Which poses the question: if financial advisers are wrong on this, what else might they be wrong about?

Apparently arid mathematical economics can therefore sometimes undermine orthodoxy and vested interests.

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